Step by step
The question
Question: graph y = 2 sin(2(x − π/4)) + 1.
1 Read A, B, C, D
Every sine graph can be written like this: y = A sin(B(x − C)) + D.
Each letter does one job.
Here A is 2, the number in front.
B is 2, the number in front of x.
x minus C: C is π/4.
D is 1, the number added at the end.
2 Find the amplitude
The amplitude is how far the wave goes up and down from its middle line.
It is |A|: 2.
3 Find the period
One full wave of sin x is a full turn: 2π long.
B squeezes the wave: the period is 2π over B.
Careful!
B squeezes the wave, so the period gets shorter.
We divide by B.
We do not multiply.
B is 2.
So the period is π: one full wave every π.
4 Find the shifts
C moves the wave sideways.
x − π/4 moves it right: the wave starts at x = π/4.
D moves the wave up 1.
Its middle line is y = 1, not the x-axis.
5 Make the key points
We use a table with five key points: one full wave.
x starts at π/4.
Each next x is a quarter of the period further: π/4.
A quarter of π is π/4.
The sin part follows the same pattern in every sine wave: 0, 1, 0, −1, 0.
Each y is A times that, plus D: times 2, then plus 1.
6 Draw the graph
Draw the grid.
The x-axis is in radians: a mark every π/4.
The middle line y = 1, dashed.
Plot the key points from the table.
Join them with a smooth wave: round at the top and the bottom, never pointed.
The wave keeps going the same way, both ways.
7 Check
Let's check one point from the table: x = π/2.
Put it in the equation.
The inside is π/2, and sin π/2 is 1.
So y is 3, like in the table.
So the graph is right!
Let's look back.
First: read A, B, C, D.
Then: find the amplitude.
Then: find the period.
Then: find the shifts.
Then: make the key points.
Then: draw the graph.
Last: check.